Finite 2-groups having a cyclic or dihedral maximal subgroup and arc-transitive maps

We classify all finite 2-groups that have a cyclic or dihedral maximal subgroup and determine their automorphism groups. Based on this result, we classify all pairs $(G,\mathcal{M})$, such that $G$ is a finite 2-group and $\mathcal{M}$ is a $G$-arc-transitive map with Euler characteristic not being divisible by 4.